#8467216

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,8)(3,9,4,6)(5,13,15,10)(11,14,12,16)(17,18,23,26)(19,24,21,20)(22,25,31,30)(27,32,29,28)(33,34,39,38)(35,40,45,43)(36,41,46,48)(37,47,44,42);(2,8)(3,4)(10,13)(11,12)(17,19,23,21)(18,20,26,24)(22,27,31,29)(25,28,30,32)(33,34,39,38)(35,43,45,40)(36,41,46,48)(37,42,44,47);(1,3,7,4)(2,6,8,9)(5,11,15,12)(10,14,13,16)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,35,39,45)(34,40,38,43)(36,44,46,37)(41,42,48,47);(1,4,7,3)(2,9,8,6)(5,12,15,11)(10,16,13,14)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,32)(28,29)(33,36)(34,41)(35,44)(37,45)(38,48)(39,46)(40,42)(43,47);(1,5,7,15)(2,10,8,13)(3,11,4,12)(6,14,9,16)(17,21,23,19)(18,24,26,20)(22,29,31,27)(25,32,30,28)(33,37,39,44)(34,42,38,47)(35,36,45,46)(40,48,43,41);(1,6,7,9)(2,3,8,4)(5,14,15,16)(10,11,13,12)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,38,39,34)(35,43,45,40)(36,41,46,48)(37,47,44,42)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=ghi,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8374412);